Structural optimization considering smallest magnitude eigenvalues: a smooth approximation

Abstract

An issue that frequently arises in structural optimization problems considering eigenvalues is the non differentiability of repeated eigenvalues. In order to overcome this difficulty, several schemes were already presented in the literature. However, these approaches generally have other disadvantages such as the inclusion of additional constraints, the inaccuracy of representation of smallest/largest eigenvalues, the significant increase in the computational effort required and incompatibility with finite differences schemes. In this paper a smooth p-norm approximation for the smallest magnitude eigenvalue is employed. The resulting approximation is differentiable, converges to the exact value as p is increased and is very simple to use (it is also compatible with finite difference schemes). Although the use of smooth approximations for maximum/minimum operators is a classical approach, for some reason it was not extensively studied in the context of structural optimization considering eigenvalues. Three examples concerning topology optimization for the maximization of the first natural vibration frequency of plane stress structures are presented in order to show the effectiveness of the proposed approach.

Appendix

Lemma 1

Take a nonsingular symmetric matrix $\mathbf{M} \in \mathbb {R}^{m \times m}$ and two $\mathbf{M}$ -orhtonormal sets $\beta = \{ {{\phi }}_{1}, {{\phi }}_{2}, \ldots , {{\phi }}_{n} \}$ and $\beta ' = \{ {{\psi }}_{1}, {{\psi }}_{2}, \ldots , {{\psi }}_{n} \}$ that span the same subspace $\mathbb {R}^{n}$ with $n\le m$. Then

$$\begin{aligned} \sum _{i=1}^{n} {{\phi }}_{i}^{T} \mathbf{A} {{\phi }}_{i} = \sum _{i=1}^{n} {{\psi }}_{i}^{T} \mathbf{A} {{\psi }}_{i}, \end{aligned}$$

(26)

for all $\mathbf{A} \in \mathbb {R}^{m \times m}$, $m \ge n$.

Proof

Since both $\beta '$ and $\beta$ span the same subspace $\mathbb {R}^{n}$ it is possible to write

$$\begin{aligned} \phi _{ij} = c_{ik} \psi _{kj}, \quad i=1,2,\ldots ,n, \end{aligned}$$

(27)

where $\phi _{ij}$ represents component j from vector i and repeated index imply summation. The $\mathbf{M}$-orthonormality condition of $\beta$ gives

$$\begin{aligned} \phi _{ik} M_{kl} \phi _{jl} = c_{im} \psi _{mk} M_{kl} c_{jn} \psi _{nl} = \delta _{ij}, \end{aligned}$$

(28)

where $\delta _{ij}$ is Kronecker’s Delta. Since the vectors from $\beta '$ are also $\mathbf{M}$-orthonormal (i.e. $\psi _{mk} M_{kl} \psi _{nl} = \delta _{mn}$) the previous expression gives

$$\begin{aligned} c_{im} c_{jn} \delta _{mn} = c_{im} c_{jm} = \delta _{ij}, \end{aligned}$$

(29)

$$\begin{aligned} c_{im} c_{jm} = c_{mi} c_{mj} = \delta _{ij}, \end{aligned}$$

(30)

(i.e. $\mathbf{C} \mathbf{C}^{T} = \mathbf{C}^{T} \mathbf{C} = \mathbf{I}$) and we conclude that $\mathbf{C}$ is orthogonal (even though the original hypothesis lied on $\mathbf{M}$-orthonormality, not orthonormality in the common sense).

The left hand side of Eq. (26) can be written as

$$\begin{aligned} \phi _{ik} A_{kl} \phi _{il} = c_{im} \psi _{mk} A_{kl} c_{in} \psi _{nl}. \end{aligned}$$

(31)

From Eq. (30) we conclude that $c_{im} c_{in} = \delta _{mn}$ and, consequently, the previous equation becomes

$$\begin{aligned} \phi _{ik} A_{kl} \phi _{il} = \delta _{mn} \psi _{mk} A_{kl} \psi _{nl}, \end{aligned}$$

(32)

$$\begin{aligned} \phi _{ik} A_{kl} \phi _{il} = \psi _{nk} A_{kl} \psi _{nl}, \end{aligned}$$

(33)

that completes the proof. $\square$

Remark 4

A particular case of the previous results follows from $\mathbf{M} = \mathbf{I}$. In this case we can choose the canonical basis $\beta ' = \{\mathbf{e}_{1},\mathbf{e}_{2},\ldots ,\mathbf{e}_{n} \}$ and some another orthonormal basis $\beta = \{{{\phi }}_{1},{{\phi }}_{2},\ldots ,{{\phi }}_{n} \}$, related by ${{\phi }}_{i} = \mathbf{C} {\varvec{e}}_{i}$, where $\mathbf{C}$ is an orthogonal matrix. From Eq. (26) we then get

$$\begin{aligned} \sum _{i=1}^{n} {\varvec{e}}_{i}^{T} \mathbf{A} {\varvec{e}}_{i} = \sum _{i=1}^{n} {\varvec{e}}_{i}^{T} \mathbf{C}^{T} \mathbf{A} \mathbf{C} {\varvec{e}}_{i}. \end{aligned}$$

(34)

Using ${{\mathrm{tr}}}( )$ to write the trace of a given matrix we conclude that

$$\begin{aligned} {{\mathrm{tr}}}(\mathbf{A}) = {{\mathrm{tr}}}(\mathbf{C}^{T} \mathbf{A} \mathbf{C}), \end{aligned}$$

(35)

that is a well known result from linear algebra. In this context, we can say that the previous lemma is a generalized version of this well known trace equality for $\mathbf{M}$-orthonormal bases.